Answer to Question #328157 in Statistics and Probability for Payal

Question

 The intelligence quotients (IQs) of 16 students from one area of a city showed a mean of 107 and a standard deviation of 10, while the IQs of 14 students from another area of the city showed a mean of 112 and a standard deviation of 8. What would be the ‘ t ’ value to test whether there is a significant difference between the IQs of the two groups at significance level 0.01?

a. 2.56

b. 1.45

c. 3.45

d. 2.22

Answer 1

$n_1=16\\n_2=14\\\bar{x}_1=107\\\bar{x}_2=112\\s_1=10\\s_2=8\\\nu =\frac{\left( \frac{{s_1}^2}{n_1}+\frac{{s_2}^2}{n_2} \right) ^2}{\frac{1}{n_1-1}\left( \frac{{s_1}^2}{n_1} \right) ^2+\frac{1}{n_2-1}\left( \frac{{s_2}^2}{n_2} \right) ^2}=\frac{\left( \frac{10^2}{16}+\frac{8^2}{14} \right) ^2}{\frac{1}{15}\left( \frac{10^2}{16} \right) ^2+\frac{1}{13}\left( \frac{8^2}{14} \right) ^2}=27.8043\approx 28\\T=\left| \frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{{s_1}^2}{n_1}+\frac{{s_2}^2}{n_2}}} \right|=\left| \frac{107-112}{\sqrt{\frac{10^2}{16}+\frac{8^2}{14}}} \right|=1.51994\\t_{\frac{1+\gamma}{2},\nu}=t_{0.995,28}=2.76\\None\,\,is\,\,correct$